Integral Tate modules and splitting of primes in torsion fields of elliptic curves

Let $E$ be an elliptic curve over a finite field $k$, and $\mathrm{\ell }$ a prime number different from the characteristic of $k$. In this paper we consider the problem of finding the structure of the Tate module $T_{\mathrm{\ell }}(E)$ as an integral Galois representations of $k$. We indicate an explicit procedure to solve this problem starting from the characteristic polynomial $f_E(x)$ and the $j$-invariant $j_E$ of $E$. Hilbert Class Polynomials of imaginary quadratic orders play here an important role. We give a global application to the study of prime-splitting in torsion fields of elliptic curves over number fields.